Impact of quantum entanglement on spectrum of cosmological fluctuations

We investigate the effect of entanglement between two causally separated open charts in de Sitter space on the spectrum of vacuum fluctuations. We consider a free massive scalar field, and construct the reduced density matrix by tracing out the vacuum state for one of the open charts, as recently derived by Maldacena and Pimentel. We formulate the mean-square vacuum fluctuations by using the reduced density matrix and show that the scale invariant spectrum of massless scalar field is realized on small scales. On the other hand, we find that the quantum entanglement affects the shape of the spectrum on large scales comparable to or greater than the curvature radius.


Introduction
Quantum entanglement is one of the most fundamental and fascinating features of quantum mechanics. The most mysterious aspect of quantum entanglement would be to affect the outcome of local measurements instantaneously beyond the lightcone once a local measurement is performed. There are many phenomena in which quantum entanglement may play a role, including the bubble nucleation [1]. Recent studies of the bubble nucleation problem infer that observer frames will be strongly correlated to each other when they observe the nucleation frame [2,3,4].
The entanglement entropy as a suitable measure of entanglement of a quantum system has been developed in condensed matter physics, quantum information and high energy physics. Especially, the entanglement entropy has now been established as a useful tool in quantum field theory to characterize the nature of long range correlations.
Even so, however, the explicit calculation of the entanglement entropy in quantum field theories had not been an easy task until Ryu and Takayanagi proposed a method of calcu-lating the entanglement entropy of a strongly coupled quantum field theory with its gravity dual using holographic techniques [5]. Their formula has passed many consistency checks and proven to be extremely powerful [6].
Following a great deal of attention paid to the success, Maldacena and Pimentel developed an explicit method to calculate the entanglement entropy in a quantum field theory in the Bunch-Davies vacuum of de Sitter space and discussed the gravitational dual of this theory and its holographic interpretation [7]. The method is also extended to α-vacua in [8,9] Based on these developments that enable us to calculate long range correlations explicitly, it would be interesting to apply this to cosmology now. We expect that the entanglement could exist beyond the Hubble horizon because de Sitter expansion eventually separates off a pair of particles created within a causally connected, Hubble horizon size region. It may be possible to investigate whether a universe entangled with our own universe exists within the multiverse framework. Such a scenario may be observable through the cosmic microwave background radiation (CMB).
In this paper, we investigate the effect of quantum entanglement on the spectrum of vacuum fluctuations in de Sitter space by using a reduced density matrix in the open chart derived by Maldacena and Pimentel as a first step of application to cosmology. It is known that the inside of a nucleated bubble looks like an open universe [1], so this formulation will be suitable for the multiverse framework and also for open inflation directly. The open inflation models are discussed extensively in [10,11,12,13,14,15].
The paper is organized as follows. In section 2, we review the spectrum of quantum fluctuations in the open chart. In section 3, we review the method to derive the reduced density matrix developed by Maldacena and Pimentel with some comments relevant to its application to cosmology. Then we discuss the spectrum of quantum fluctuations using the reduced density matrix. We show the scale invariant spectrum is realized. In section 4, we find the effect of the entanglement appears in the spectrum at large wavelengths comparable to the curvature radius. Our results are summarized and discussed in section 5.

Mode functions in the open chart
The open de Sitter space is studied in detail in [10]. In Figure 1, the de Sitter space and the Penrose diagram is depicted 1 . We consider a free scalar field of mass m in de Sitter space with the action given by The metric in each region R and L may be obtained by analytic continuation from an Eulidean four-sphere metric and expressed, respectively, as where dΩ 2 is the metric on the two-sphere. Since the R and L regions are completely symmetric, if we perform the separation of variables the equations of motion for χ p and Y p m in the R or L regions are found to be in common where (t, r) = (t R , r R ) or (t L , r L ), L 2 is the Laplacian operator on the unit two-sphere, and Y p m are harmonic functions on the three-dimensional hyperbolic space. We consider the positive frequency mode functions corresponding to the Euclidean vacuum (the Bunch-Davies vacuum [16,17,18]), because it is natural that the initial state is the de Sitter invariant vacuum. They are found explicitly in [10] and the time dependent part of χ p (t) is given by are the associated Legendre functions, and the index σ takes the values ±1 and distinguishes two independent solutions for each region. We have defined a parameter The above is a solution supported both on the R and L regions. The factor e −πp in the above solutions comes from the requirement of analyticity in the Euclidean hemisphere which selects the Bunch-Davies vacuum.
We expand the field in terms of the creation and anihilation operators, φ(t, r, Ω) = dp Without loss of generality, we assume the normalization of Y p m is such that Y * p m = Y p −m . Then Eq. (2.8) may be written aŝ where we introduced a Fourier mode field operator For convenience, we write the associated Legendre functions of the R and L regions in a simple form P R,L ≡ P ip ν−1/2 (cosh t R,L ) , P R * ,L * ≡ P −ip ν−1/2 (cosh t R,L ), then the two lines of Eq. (2.6) may be expressed in one line as where the index p of χ p,σ is omitted and we defined , (2.14) and N p is a normalization factor including the 1/(2 sinh πp) in Eq. (2.6), given by In the above and in what follows, it is understood that the function P q (q = R or L) defined only in the q region is associated with a step function which is unity in the q region and which vanishes in the opposite region. The complex conjugate of Eq. (2.12) which is necessary in Eq. (2.10) is If we introduce a 4 × 4 matrix where σ = ±1, q = (R, L) and the summation is understood when a pair of the same indices appear upstairs and downstairs, then both of Eqs. (2.12) and (2.16) can be accommodated into the simple matrix form, where the capital indices (I, J) run from 1 to 4. The Fourier mode field operator Eq. (2.11) is then expanded as where t stands for a time slice given by t = t R in the R region and t = t L in the L region, and in the following the indices p, , m of φ p m (t) are omitted for simplicity unless there may be any confusion.

Spectrum in the L region
Let us calculate the power spectrum, say, in the L region because the R and L regions are completely symmetric. As it is unnecessary to consider the R region, the mode function χ I given by Eq. where The Fourier mode field operator in Eq. (2.19) is now expressed as where we used Eq. (2.20).
If we focus on a single mode with indices p, , m, the mean-square vacuum fluctuation is then computed by using Eq.
Then the normalized spectrum per unit logarithmic interval of p is given by where χ σ = χ p,σ (t L ).

Vacuum fluctuations after horizon exit
In this subsection, we make sure that the amplitude of the vacuum fluctuation is frozen then |χ σ | 2 in the normalized spectrum in Eq. (2.24) is expressed as Since the Legendre functions on superhorizon scale behave as we find the time dependence of each combination of the associated Legendre functions in Eq. (2.26) is the same. Then σ=±1 |χ σ | 2 is given by where we defined the time independent part of σ=±1 |χ σ | 2 by Z.
Plugging Eq. (2.28) into Eq. (2.24), we find the spectrum approaches For massless case ν = 3/2, we find the time dependent part becomes Thus, the vacuum fluctuation gets frozen out after horizon exit.

Wavenumber dependence
We also need to check the behavior of the spectrum at short wavelengths (p 1). The spectrum should be the same as the case of a specially flat universe. For the massless scalar Then the time independent part of Z at short wavelengths (p 1) becomes Thus the spectrum of the massless scalar field after horizon exit is evaluated as (2.33) This is the well-known result for vacuum fluctuations after horizon exit. The spectrum as a function of p is plotted in Figure 2. 3 Spectrum of quantum fluctuations using the reduced density matrix In previous section, we discussed the spectrum of vacuum fluctuations by using the field operator expanded in terms of the mode functions defined for the Bunch-Davies vacuum.
We next want to explore the spectrum of vacuum fluctuations by using the reduced density matrix derived by Maldacena and Pimentel in [7]. In this section, we first review the formalism to obtain the reduced density matrix and then discuss the spectrum of vacuum fluctuations.

A review on the reduced density matrix
In order to derive the reduced density matrix in the region L, we need to trace over the degrees of freedom of the region R. To make this procedure practically possible, the reduced density matrix has to be diagonalized. The following is a review on the derivation of the reduced density matrix given in [7].
As we see in Eq. (2.19), the mode function χ I corresponding to the Bunch-Davies vacuum is a linear combination of the Legendre functions P R,L = P ip ν−1/2 (cosh t R,L ), which are proportional to the positive frequency modes in the past in each R or L region. Setting τ = cosh t, the Legendre functions are known to be normalized as where q = R or L. Thus, the properly normalized positive frequency functions in each R or L region are given by Let us introduce ϕ I as where m ij is a symmetric matrix and the operators b i satisfy the commutation relation Note that the normalization of the Bogoliubov transformation is omitted here for simplicity because it is unnecessary for the derivation of the reduced density matrix, which will be given in Eq. (3.17).
Here we mention a subtlety associated with the fact that the p-mode spectrum is continuous. Since the commutation relation between the p-mode annihilation operator and the p -mode creation operator is proportional to δ(p − p ), that between the same spectral index p = p would diverge. To avoid this divergence, we discretize the p-mode spectrum with a width ∆p and take the limit ∆p → 0 only at the end of computation. This means, for example, we rescale the operators b i as , where the index 'cont' denotes the operator with the original, continuous spectrum, and the operators b i and b j appearing in Eq. (3.6) should be understood as the rescaled ones.
The condition a σ |BD = 0 determines m ij : where e iθ contains all unimportant phase factors for ν 2 > 0. When the state is written in the form of Eq.
to obtain the relation Here, the normalizaion |u| 2 − |v| 2 = 1 and |ū| 2 − |v| 2 = 1 are assumed so that the new operators satisfy the commutation relation [c i , c † j ] = δ ij . The normalization factor N γp is given by where |γ p | < 1 is imposed. Eq. (3.8) is a linear transformation between c q and b q , so this Bogoliubov transformation does not mix R and L Hilbert spaces, but the basis vacuum changes from |R |L to |R |L .
For later convenience, we introduce a 4 × 4 matrix form of Eq. (3.8), The consistency conditions for Eq. (3.9) are where ω * = ω and ζ * = −ζ. We see that setting v * =v and u * =ū is possible if γ p is pure imaginary γ * p = −γ p . This is always possible by adjusting the phase of c q . Then Eq. (3.14) becomes identical with Eq. (3.13) and the system is reduced to that of two homogeneous equations. We look for such γ p , keeping the normalization condition |u| 2 − |v| 2 = 1 satisfied.
In order to have a non-trivial solution in the system of equations (3.13), γ p must be where we took a minus sign in front of the square root term to satisfy |γ p | < 1. Note that γ p is pure imaginary. Putting the ω and ζ defined in Eq. (3.7) into Eq. (3.15), we get cosh 2πp + cos 2πν + √ cosh 2πp + cos 2πν + 2 . (3.16) Now we have the density matrix which enables us to trace over the R degrees of freedom easily. By using Eqs. (3.9) and (3.10), the reduced density matrix is then found to be diagonalized as |γ p | 2n |n; p m n; p m| , where we defined |n; p m = 1/ √ n! (c † L ) n |L . Note that this density matrix is for each mode labeled by p, , m.

Spectrum of vacuum fluctuations
Now we calculate the mean-square vacuum fluctuations by using the reduced density matrix obtained in Eq. (3.17). Since we traced out the degrees of freedom of the R region, the corresponding field operator has to be defined only in the L region. Also the field operator should act only on |L conforming with the reduced density matrix. Then, the Fourier mode field operator should be defined as where the calligraphic indices (I, J ) run from 1 to 2, and the matrix is reduced to a 2 × 2 form: Note that we transformed the field operator expanded by b J in Eq. (3.4) into c I by using Eq. (3.11) 2 .
Focusing on a single mode with indices p, , m, the mean-square vacuum fluctuations are then calculated as

Spectrum in the L region
We calculate the power spectrum in the L region. The mode function given in Eq. (3.18) is written as where we defined Here,ū,v are obtained by solving Eq. (3.13) with the solution Eq. (3.16) and imposing the normalization condition |ū| 2 − |v| 2 = 1: Here, ω = m RR = m LL and ζ = m LR = m RL in Eq. (3.7). Using these we can compute the expectation value for each n-particle state in Eq. (3.20), where from the second line to the third line we used L |(c L ) n c I ψ I c J ψ J † (c † L ) n |L = |ψ L | 2 n! n; p m|n; p m + 2|ψ L | 2 n 2 (n − 1)! n − 1; p m|n − 1; p m = (2n + 1) n! |ψ L | 2 .
(3.25) Putting Eq. (3.24) into Eq. (3.20), the mean-square vacuum fluctuations for each mode is given by Note that by comparing with Eq. (2.23), we see the main effect of tracing out the degrees of freedom of the region R seems to come in with the form of (1 + |γ p | 2 )/(1 − |γ p | 2 ). Since |γ p | → 1 as p → 0 (for m = 0), this extra term enhances the mean-square vacuum fluctuations at long wavelengths (p 1). Note also that the mode functions are changed from

Vacuum fluctuations after horizon exit
Since the Legendre functions on superhorizon scale is the same as in Eq. (2.27), we find the time dependence of this case is the same as Eq. (2.28) and expressed as where we defined Z corresponding to Z in Eq. (2.28) for comparison. The spectrum of the vacuum fluctuations corresponding to Eq. (2.24) is now given by Plugging Eqs. (3.26) and (3.29) into above, we find the spectrum is found to be We see the time dependent part is completely identical with Eq. (2.29). Thus, for massless case (ν = 3/2), the vacuum fluctuation gets frozen after horizon exit in the case of the spectrum using the reduced density matrix as well.

Wavenumber dependence
Next, let us see the behavior of the power spectrum at short wavelengths (p 1) as we did in subsection 2.4. For a massless scalar field (ν = 3/2), the dominant term in Z of Eq. (3.29) for large p is found to be (3.32) The corresponding case of the Bunch-Davies vacuum is Eq. (2.31). We see the α σ L /N p is simply replaced byū/N b , and find that the behavior of Z is the same as that of Z, (3.33) Thus the spectrum of a massless scalar field after horizon exit is evaluated as We find that the scale invariant spectrum is realized for p 1 even after tracing out the degrees of freedom of the region R. We plot the spectrum of the massless scalar field in Eq. (3.34) as a function of p in Figure 3. It is almost identical to Figure 2.

Possible observable signatures
So far, we discussed two different spectra given in terms of χ I and ψ I . The former correspond to the mode functions for the entangled Bunch-Davies vacuum, and the latter to those that diagonalize the reduced density matrix obtained from the Bunch-Davies vacuum by tracing out the degrees of freedom of the R region. The corresponding states are given by the density matrix ρ BD = |BD BD| and the reduced density matrix ρ L = Tr R ρ BD , respectively.
The difference between these two states is the observer's point of view when taking the mean-square vacuum fluctuations. For ρ BD , we are supposed to know the entangled state in the region R. On the other hand, for ρ L , we are supposed to be completely ignorant about the state in the R region. In order to distinguish them, we call ρ BD the "entangled state", and ρ L the "mixed state (obtained from the entangled state)", or just denote them by their relevant mode functions χ I and ψ I in the following.

Non-entangled state
In the subsections 2.4 and 3.5, we showed that both of the entangled state and the mixed state realize the scale invariant spectrum for a massless scalar field at short wavelengths (p 1). Now we wonder what would happen to the spectrum if we assume the L region were the whole universe and use the mode function ϕ I which is defined only in the L region.
We call it "non-entangled state", and denote it by ϕ I .
The relevant mode function for the non-entangled case should be given by Eq. (3.2).
That is The spectrum is then defined by where the Fourier mode field operator is and we find the mean-square vacuum fluctuations for each mode The |ϕ L | 2 becomes a simple form Here, time independent part ofZ at short wavelengths (p 1) is evaluated as The power spectrum becomes (4.7) Thus, for p 1, we get the flat spectrum of vacuum fluctuations for a massless scalar field even for the non-entangled state. This means that we cannot distinguish which state is valid for our universe as far as we focus on the short wavelength region of the spectrum. So let us examine the behavior at long wavelengths after the horizon exit.

Spectra at long wavelengths
In order to see the behavior at long wavelengths, we take the limit p 1 and expand the expressions for the spectra in p. For ϕ I and χ I , we expandZ in Eq. (4.5) and Z in Eq. (2.28), respectively. Then we find the p dependence at leading order is given by  In the case of ψ I , we expand Z in Eq. (3.29) to obtain P(p) We are interested in the case of small mass (|ν − 3/2| 1). Taking this limit, we see that the spectrum for ψ I decreases most slowly as p → 0. The spectra at long wavelengths are plotted in Figure 4, where we take m 2 = H 2 /10. The spectra for the entangled state (χ I ) and the mixed state (ψ I ) are the blue line and the red line, respectively. The green line is the spectrum for the non-entangled state (ϕ I

Summary and discussion
We investigated the spectrum of vacuum fluctuations with quantum entanglement in de Sitter space by using the reduced density matrix obtained by Maldacena and Pimentel in [7]. We formulated the mean-square vacuum fluctuations by introducing the two open charts, L and R, which are entangled. We found that the effect of tracing out the degrees of freedom of the R region comes in to the mean-square of vacuum fluctuations and enhances it on scales larger than the curvature radius. On the other hand, on small scales, the spectrum of vacuum fluctuations of a massless scalar field is found to be scale invariant, and completely indistinguishable from the pure Bunch-Davies (hence highly entangled) case.
We also considered a fictitious case in which we assume the L region were the whole universe. In this case, with respect to a natural vacuum state defined in the L region, the scale invariant spectrum is also realized on small scales. Thus it turned out that we cannot distinguish which state is valid for our universe among the entangled state (χ I ), the mixed state from the entangled state (ψ I ) and non-entangled state (ϕ I ), as long as we focus on the spectrum on small scales.
On the other hand, the spectrum for each of these three cases is found to be different from each other on large scales comparable to or greater than the curvature radius.
We found that the spectra for the entangled and mixed states tend to be enhanced compared to the non-entangled state on scales comparable to the curvature radius. If we go further to scales much larger than the curvature radius, there appears a difference between the entangled state and mixed state. The spectrum for the mixed state is enhanced substantially relative to the entangled state. Since we can use these differences to distinguish those three states, they seem to be relevant (and hopefully observationally testable) in the context of open inflation. Now let us consider why the difference appeared in the spectra between the entangled and mixed states. In general, if we consider two subsystems A and B that form a state |Ψ , the expectation value of a physical quantity O is expressed by using the density matrix Based on this general argument, we may understand the reason why the spectra for the entangled state and the mixed state from the entangled state become different on large scale (p → 0). We see that the creation and anihilation operators a J include the operators on both sides, b J = (b q , b † q ) (q = R, L), as in Eq. (3.5). This means that the field operator φ(t) naturally involves operators on both sides even if the time slice t is restricted to the L region. On the other hand, after tracing out the degrees of freedom of the R region, the mixed state (ψ I ) is described by the creation and anihilation operators defined only in the L region. That is, the field operator φ L (t) is defined only in the L region. Thus apparently φ(t) = φ L (t), implying that the above general argument does not hold because O = O A .
Finally, we mention that it would be easy to extend our analysis to the case of gravitons.
Another direction is to consider interactions. The formalism we developed in this paper is at tree level. It would be interesting to consider the loop corrections and see how they affect the result. They might enhance the effect of the entanglement on small scales because of the coupling between long and short wavelengths [19].